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 3 times and lost 7 times.

1. Goat
2. Car
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
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 324 times and lost 676 times.

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