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 6 times and lost 4 times.

1. Car
2. Goat
3. Goat
4. Car
5. Car
6. Car
7. Car
8. Goat
9. Car
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 321 times and lost 679 times.

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