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. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Goat
10. Car

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 366 times and lost 634 times.

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