Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 1 time and lost 9 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 290 times and lost 710 times.

1. Car
2. Car
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat
11. Car
12. Car
13. Goat
14. Goat
15. Goat
16. Goat
17. Goat
18. Goat
19. Goat
20. Car
21. Car
22. Car
23. Goat
24. Goat
25. Goat
26. Car
27. Goat
28. Goat
29. Goat
30. Goat
31. Goat
32. Car
33. Goat
34. Car
35. Car
36. Goat
37. Goat
38. Car
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Car
45. Goat
46. Goat
47. Car
48. Goat
49. Goat
50. Goat
51. Car
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Car
58. Car
59. Goat
60. Goat
61. Goat
62. Car
63. Goat
64. Car
65. Goat
66. Goat
67. Goat
68. Goat
69. Car
70. Goat
71. Goat
72. Car
73. Goat
74. Goat
75. Car
76. Goat
77. Car
78. Goat
79. Goat
80. Car
81. Goat
82. Goat
83. Goat
84. Goat
85. Goat
86. Goat
87. Car
88. Car
89. Goat
90. Goat
91. Goat
92. Goat
93. Goat
94. Goat
95. Goat
96. Goat
97. Goat
98. Goat
99. Goat
100. Goat
101. Car
102. Car
103. Car
104. Car
105. Goat
106. Goat
107. Goat
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Car
115. Car
116. Car
117. Car
118. Goat
119. Goat
120. Car
121. Goat
122. Goat
123. Goat
124. Goat
125. Goat
126. Goat
127. Goat
128. Car
129. Car
130. Goat
131. Goat
132. Goat
133. Goat
134. Goat
135. Car
136. Goat
137. Car
138. Goat
139. Car
140. Car
141. Goat
142. Goat
143. Goat
144. Goat
145. Goat
146. Goat
147. Car
148. Goat
149. Goat
150. Goat
151. Car
152. Goat
153. Goat
154. Goat
155. Goat
156. Goat
157. Goat
158. Car
159. Goat
160. Goat
161. Goat
162. Goat
163. Goat
164. Goat
165. Goat
166. Goat
167. Goat
168. Car
169. Goat
170. Goat
171. Goat
172. Goat
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Goat
179. Goat
180. Goat
181. Goat
182. Goat
183. Car
184. Goat
185. Goat
186. Car
187. Goat
188. Goat
189. Goat
190. Goat
191. Car
192. Car
193. Car
194. Goat
195. Car
196. Goat
197. Goat
198. Goat
199. Goat
200. Goat
201. Car
202. Goat
203. Goat
204. Car
205. Goat
206. Goat
207. Goat
208. Goat
209. Car
210. Goat
211. Goat
212. Car
213. Goat
214. Car
215. Goat
216. Goat
217. Goat
218. Goat
219. Goat
220. Car
221. Car
222. Car
223. Car
224. Goat
225. Car
226. Goat
227. Goat
228. Goat
229. Car
230. Goat
231. Goat
232. Goat
233. Goat
234. Goat
235. Car
236. Car
237. Car
238. Goat
239. Goat
240. Car
241. Goat
242. Goat
243. Car
244. Car
245. Car
246. Goat
247. Car
248. Goat
249. Goat
250. Car
251. Goat
252. Car
253. Goat
254. Goat
255. Goat
256. Goat
257. Goat
258. Goat
259. Car
260. Car
261. Car
262. Car
263. Goat
264. Car
265. Goat
266. Goat
267. Car
268. Goat
269. Car
270. Goat
271. Goat
272. Goat
273. Car
274. Goat
275. Car
276. Goat
277. Car
278. Goat
279. Car
280. Goat
281. Goat
282. Goat
283. Goat
284. Car
285. Goat
286. Goat
287. Car
288. Goat
289. Goat
290. Goat
291. Car
292. Goat
293. Goat
294. Car
295. Goat
296. Car
297. Car
298. Goat
299. Goat
300. Goat
301. Car
302. Goat
303. Goat
304. Goat
305. Car
306. Goat
307. Car
308. Goat
309. Goat
310. Car
311. Car
312. Car
313. Goat
314. Goat
315. Goat
316. Goat
317. Goat
318. Car
319. Goat
320. Goat
321. Car
322. Car
323. Goat
324. Goat
325. Car
326. Car
327. Goat
328. Goat
329. Goat
330. Car
331. Goat
332. Goat
333. Car
334. Car
335. Goat
336. Goat
337. Goat
338. Car
339. Car
340. Goat
341. Goat
342. Goat
343. Car
344. Goat
345. Goat
346. Goat
347. Goat
348. Goat
349. Car
350. Goat
351. Goat
352. Goat
353. Goat
354. Goat
355. Car
356. Goat
357. Goat
358. Goat
359. Goat
360. Car
361. Car
362. Car
363. Car
364. Goat
365. Goat
366. Goat
367. Goat
368. Goat
369. Car
370. Goat
371. Car
372. Car
373. Goat
374. Goat
375. Goat
376. Car
377. Car
378. Car
379. Car
380. Goat
381. Goat
382. Goat
383. Car
384. Car
385. Car
386. Goat
387. Car
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Goat
394. Car
395. Car
396. Car
397. Goat
398. Goat
399. Goat
400. Car
401. Goat
402. Car
403. Car
404. Car
405. Car
406. Goat
407. Goat
408. Goat
409. Goat
410. Goat
411. Goat
412. Car
413. Goat
414. Goat
415. Car
416. Goat
417. Goat
418. Car
419. Goat
420. Car
421. Goat
422. Goat
423. Goat
424. Goat
425. Car
426. Goat
427. Goat
428. Goat
429. Car
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Car
436. Goat
437. Car
438. Goat
439. Goat
440. Goat
441. Car
442. Goat
443. Car
444. Goat
445. Goat
446. Goat
447. Car
448. Car
449. Goat
450. Car
451. Goat
452. Car
453. Car
454. Car
455. Goat
456. Goat
457. Goat
458. Car
459. Car
460. Goat
461. Goat
462. Goat
463. Goat
464. Car
465. Goat
466. Goat
467. Car
468. Goat
469. Car
470. Car
471. Car
472. Goat
473. Goat
474. Goat
475. Goat
476. Goat
477. Goat
478. Car
479. Car
480. Goat
481. Goat
482. Car
483. Goat
484. Goat
485. Goat
486. Goat
487. Goat
488. Car
489. Car
490. Goat
491. Goat
492. Goat
493. Goat
494. Goat
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Goat
501. Car
502. Goat
503. Goat
504. Goat
505. Car
506. Goat
507. Goat
508. Goat
509. Goat
510. Goat
511. Car
512. Goat
513. Goat
514. Goat
515. Goat
516. Car
517. Goat
518. Goat
519. Goat
520. Goat
521. Goat
522. Goat
523. Goat
524. Goat
525. Car
526. Goat
527. Goat
528. Car
529. Goat
530. Goat
531. Goat
532. Goat
533. Goat
534. Car
535. Goat
536. Car
537. Car
538. Goat
539. Car
540. Goat
541. Goat
542. Car
543. Goat
544. Goat
545. Goat
546. Car
547. Goat
548. Goat
549. Goat
550. Goat
551. Car
552. Goat
553. Car
554. Goat
555. Goat
556. Goat
557. Goat
558. Goat
559. Goat
560. Goat
561. Goat
562. Goat
563. Goat
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Goat
570. Goat
571. Goat
572. Car
573. Goat
574. Goat
575. Goat
576. Goat
577. Car
578. Goat
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Car
585. Goat
586. Goat
587. Car
588. Goat
589. Car
590. Goat
591. Goat
592. Goat
593. Goat
594. Car
595. Goat
596. Goat
597. Goat
598. Goat
599. Goat
600. Goat
601. Car
602. Goat
603. Goat
604. Goat
605. Goat
606. Goat
607. Goat
608. Car
609. Goat
610. Goat
611. Goat
612. Goat
613. Car
614. Goat
615. Goat
616. Goat
617. Car
618. Car
619. Goat
620. Goat
621. Goat
622. Goat
623. Goat
624. Goat
625. Goat
626. Goat
627. Car
628. Car
629. Car
630. Car
631. Car
632. Car
633. Goat
634. Car
635. Goat
636. Goat
637. Car
638. Goat
639. Goat
640. Goat
641. Car
642. Goat
643. Goat
644. Goat
645. Car
646. Goat
647. Goat
648. Goat
649. Car
650. Goat
651. Goat
652. Car
653. Goat
654. Goat
655. Goat
656. Goat
657. Goat
658. Goat
659. Goat
660. Goat
661. Goat
662. Goat
663. Goat
664. Goat
665. Goat
666. Car
667. Goat
668. Goat
669. Goat
670. Goat
671. Car
672. Goat
673. Goat
674. Goat
675. Car
676. Car
677. Goat
678. Goat
679. Goat
680. Goat
681. Goat
682. Goat
683. Goat
684. Goat
685. Goat
686. Car
687. Car
688. Goat
689. Goat
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Goat
697. Goat
698. Car
699. Car
700. Goat
701. Goat
702. Goat
703. Goat
704. Goat
705. Car
706. Car
707. Goat
708. Goat
709. Goat
710. Goat
711. Goat
712. Car
713. Goat
714. Goat
715. Goat
716. Goat
717. Car
718. Goat
719. Goat
720. Goat
721. Car
722. Goat
723. Goat
724. Car
725. Car
726. Goat
727. Goat
728. Car
729. Car
730. Car
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Car
737. Goat
738. Car
739. Goat
740. Car
741. Goat
742. Goat
743. Goat
744. Goat
745. Goat
746. Goat
747. Goat
748. Goat
749. Goat
750. Car
751. Goat
752. Car
753. Goat
754. Car
755. Goat
756. Goat
757. Goat
758. Car
759. Goat
760. Car
761. Car
762. Goat
763. Car
764. Car
765. Goat
766. Goat
767. Car
768. Goat
769. Goat
770. Goat
771. Goat
772. Car
773. Goat
774. Goat
775. Goat
776. Car
777. Goat
778. Goat
779. Goat
780. Goat
781. Car
782. Car
783. Car
784. Car
785. Goat
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Goat
792. Car
793. Goat
794. Goat
795. Goat
796. Goat
797. Car
798. Goat
799. Goat
800. Car
801. Goat
802. Goat
803. Goat
804. Car
805. Goat
806. Goat
807. Goat
808. Goat
809. Goat
810. Goat
811. Goat
812. Car
813. Goat
814. Car
815. Goat
816. Car
817. Goat
818. Car
819. Goat
820. Goat
821. Car
822. Car
823. Goat
824. Car
825. Car
826. Car
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Goat
835. Goat
836. Goat
837. Goat
838. Car
839. Goat
840. Goat
841. Goat
842. Goat
843. Goat
844. Car
845. Goat
846. Goat
847. Car
848. Goat
849. Goat
850. Goat
851. Goat
852. Goat
853. Goat
854. Goat
855. Car
856. Goat
857. Goat
858. Car
859. Goat
860. Car
861. Car
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Car
870. Goat
871. Goat
872. Goat
873. Goat
874. Goat
875. Car
876. Goat
877. Goat
878. Goat
879. Goat
880. Goat
881. Goat
882. Car
883. Goat
884. Goat
885. Goat
886. Goat
887. Car
888. Car
889. Car
890. Goat
891. Goat
892. Goat
893. Goat
894. Goat
895. Goat
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Car
903. Goat
904. Car
905. Goat
906. Goat
907. Car
908. Goat
909. Car
910. Goat
911. Goat
912. Car
913. Goat
914. Car
915. Car
916. Goat
917. Goat
918. Goat
919. Goat
920. Goat
921. Car
922. Car
923. Car
924. Goat
925. Goat
926. Goat
927. Car
928. Goat
929. Goat
930. Goat
931. Car
932. Goat
933. Goat
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Goat
940. Goat
941. Car
942. Car
943. Car
944. Goat
945. Goat
946. Goat
947. Goat
948. Goat
949. Goat
950. Goat
951. Goat
952. Car
953. Goat
954. Car
955. Goat
956. Goat
957. Goat
958. Car
959. Car
960. Goat
961. Car
962. Car
963. Car
964. Goat
965. Car
966. Goat
967. Car
968. Goat
969. Car
970. Goat
971. Car
972. Car
973. Goat
974. Car
975. Car
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Goat
985. Goat
986. Goat
987. Goat
988. Goat
989. Goat
990. Goat
991. Goat
992. Goat
993. Car
994. Goat
995. Goat
996. Goat
997. Goat
998. Car
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 111 times and lost 889 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 36 times and lost 64 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}